Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G

Percentage Accurate: 85.2% → 98.9%

Time: 9.9s

Alternatives: 7

Speedup: 17.6×

Specification

Math

\[\begin{array}{l} \\ x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (exp((y * log((y / (z + y))))) / y)
end function

Java

public static double code(double x, double y, double z) {
	return x + (Math.exp((y * Math.log((y / (z + y))))) / y);
}

Python

def code(x, y, z):
	return x + (math.exp((y * math.log((y / (z + y))))) / y)

Julia

function code(x, y, z)
	return Float64(x + Float64(exp(Float64(y * log(Float64(y / Float64(z + y))))) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (exp((y * log((y / (z + y))))) / y);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Exp[N[(y * N[Log[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (exp((y * log((y / (z + y))))) / y)
end function

Java

public static double code(double x, double y, double z) {
	return x + (Math.exp((y * Math.log((y / (z + y))))) / y);
}

Python

def code(x, y, z):
	return x + (math.exp((y * math.log((y / (z + y))))) / y)

Julia

function code(x, y, z)
	return Float64(x + Float64(exp(Float64(y * log(Float64(y / Float64(z + y))))) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (exp((y * log((y / (z + y))))) / y);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Exp[N[(y * N[Log[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{e^{0 - z}}{y}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (exp (- 0.0 z)) y))))
   (if (<= y -1.4e+21) t_0 (if (<= y 2e-13) (+ x (/ 1.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (exp((0.0 - z)) / y);
	double tmp;
	if (y <= -1.4e+21) {
		tmp = t_0;
	} else if (y <= 2e-13) {
		tmp = x + (1.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (exp((0.0d0 - z)) / y)
    if (y <= (-1.4d+21)) then
        tmp = t_0
    else if (y <= 2d-13) then
        tmp = x + (1.0d0 / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (Math.exp((0.0 - z)) / y);
	double tmp;
	if (y <= -1.4e+21) {
		tmp = t_0;
	} else if (y <= 2e-13) {
		tmp = x + (1.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (math.exp((0.0 - z)) / y)
	tmp = 0
	if y <= -1.4e+21:
		tmp = t_0
	elif y <= 2e-13:
		tmp = x + (1.0 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(exp(Float64(0.0 - z)) / y))
	tmp = 0.0
	if (y <= -1.4e+21)
		tmp = t_0;
	elseif (y <= 2e-13)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (exp((0.0 - z)) / y);
	tmp = 0.0;
	if (y <= -1.4e+21)
		tmp = t_0;
	elseif (y <= 2e-13)
		tmp = x + (1.0 / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(N[Exp[N[(0.0 - z), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4e+21], t$95$0, If[LessEqual[y, 2e-13], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e21 or 2.0000000000000001e-13 < y

   1. Initial program 90.5%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 90.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 90.5%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x + \frac{e^{-1 \cdot z}}{y}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{-1 \cdot z}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{-1 \cdot z}\right), \color{blue}{y}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot z\right)\right), y\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right), y\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - z\right)\right), y\right)\right) \]

      6. --lowering--.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right), y\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{e^{0 - z}}{y}} \]

   if -1.4e21 < y < 2.0000000000000001e-13

   1. Initial program 85.7%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 85.7%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

      2. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -20:\\ \;\;\;\;\frac{1}{y \cdot e^{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -20.0) (/ 1.0 (* y (exp z))) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -20.0) {
		tmp = 1.0 / (y * exp(z));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-20.0d0)) then
        tmp = 1.0d0 / (y * exp(z))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -20.0) {
		tmp = 1.0 / (y * Math.exp(z));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -20.0:
		tmp = 1.0 / (y * math.exp(z))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -20.0)
		tmp = Float64(1.0 / Float64(y * exp(z)));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -20.0)
		tmp = 1.0 / (y * exp(z));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -20.0], N[(1.0 / N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -20

   1. Initial program 54.6%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 54.6%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 54.6%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x + \frac{e^{-1 \cdot z}}{y}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{-1 \cdot z}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{-1 \cdot z}\right), \color{blue}{y}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot z\right)\right), y\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right), y\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - z\right)\right), y\right)\right) \]

      6. --lowering--.f64 61.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right), y\right)\right) \]

   7. Simplified 61.8%

      \[\leadsto \color{blue}{x + \frac{e^{0 - z}}{y}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{e^{\mathsf{neg}\left(z\right)}}{y}} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{\frac{1}{e^{z}}}{y} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot e^{z}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(y \cdot e^{z}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(e^{z}\right)}\right)\right) \]

      5. exp-lowering-exp.f64 61.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{exp.f64}\left(z\right)\right)\right) \]

   10. Simplified 61.8%

       \[\leadsto \color{blue}{\frac{1}{y \cdot e^{z}}} \]

   if -20 < z

   1. Initial program 95.2%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 95.2%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

      2. /-lowering-/.f64 95.2%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   7. Simplified 95.2%

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.5% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+166}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(z \cdot -0.16666666666666666\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.5e+166)
   (* z (/ (* z (* z -0.16666666666666666)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.5e+166) {
		tmp = z * ((z * (z * -0.16666666666666666)) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6.5d+166)) then
        tmp = z * ((z * (z * (-0.16666666666666666d0))) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.5e+166) {
		tmp = z * ((z * (z * -0.16666666666666666)) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.5e+166:
		tmp = z * ((z * (z * -0.16666666666666666)) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.5e+166)
		tmp = Float64(z * Float64(Float64(z * Float64(z * -0.16666666666666666)) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.5e+166)
		tmp = z * ((z * (z * -0.16666666666666666)) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.5e+166], N[(z * N[(N[(z * N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5000000000000005e166

   1. Initial program 62.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 62.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right)\right)}, y\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(z \cdot \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right)\right)\right), y\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right)\right)\right), y\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), y\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) + -1\right)\right)\right), y\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(-1 + z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right)\right), y\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right)\right)\right), y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right)\right)\right), y\right)\right) \]

   7. Simplified 73.7%

      \[\leadsto x + \frac{\color{blue}{1 + z \cdot \left(-1 + z \cdot \left(0.5 + \left(\frac{0.5}{y} - z \cdot \left(\frac{0.5}{y} + \left(0.16666666666666666 + \frac{0.3333333333333333}{y \cdot y}\right)\right)\right)\right)\right)}}{y} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \left({z}^{3} \cdot \left(\frac{1}{6} \cdot \frac{1}{y} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{3}} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot {z}^{3}\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{y} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{3}} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{6} \cdot \frac{1}{y} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{3}} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right) \cdot \color{blue}{\left(-1 \cdot {z}^{3}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \frac{1}{y} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{3}} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right), \color{blue}{\left(-1 \cdot {z}^{3}\right)}\right) \]

   10. Simplified 72.6%

       \[\leadsto \color{blue}{\left(\frac{0.5}{y \cdot y} + \left(\frac{0.16666666666666666}{y} + \frac{0.3333333333333333}{y \cdot \left(y \cdot y\right)}\right)\right) \cdot \left(0 - z \cdot \left(z \cdot z\right)\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{{z}^{3}}{y}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot {z}^{3}}{\color{blue}{y}} \]

       2. *-commutative N/A

          \[\leadsto \frac{{z}^{3} \cdot \frac{-1}{6}}{y} \]

       3. cube-mult N/A

          \[\leadsto \frac{\left(z \cdot \left(z \cdot z\right)\right) \cdot \frac{-1}{6}}{y} \]

       4. unpow2 N/A

          \[\leadsto \frac{\left(z \cdot {z}^{2}\right) \cdot \frac{-1}{6}}{y} \]

       5. associate-*l* N/A

          \[\leadsto \frac{z \cdot \left({z}^{2} \cdot \frac{-1}{6}\right)}{y} \]

       6. associate-/l* N/A

          \[\leadsto z \cdot \color{blue}{\frac{{z}^{2} \cdot \frac{-1}{6}}{y}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{{z}^{2} \cdot \frac{-1}{6}}{y}\right)}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left({z}^{2} \cdot \frac{-1}{6}\right), \color{blue}{y}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(\left(z \cdot z\right) \cdot \frac{-1}{6}\right), y\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \left(z \cdot \frac{-1}{6}\right)\right), y\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \left(\frac{-1}{6} \cdot z\right)\right), y\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{-1}{6} \cdot z\right)\right), y\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot \frac{-1}{6}\right)\right), y\right)\right) \]

       14. *-lowering-*.f64 73.7%

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right), y\right)\right) \]

   13. Simplified 73.7%

       \[\leadsto \color{blue}{z \cdot \frac{z \cdot \left(z \cdot -0.16666666666666666\right)}{y}} \]

   if -6.5000000000000005e166 < z

   1. Initial program 90.4%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 90.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 90.4%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

      2. /-lowering-/.f64 90.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   7. Simplified 90.8%

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 64.7% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+151}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-60}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.55e+151) x (if (<= y 1.55e-60) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e+151) {
		tmp = x;
	} else if (y <= 1.55e-60) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.55d+151)) then
        tmp = x
    else if (y <= 1.55d-60) then
        tmp = 1.0d0 / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e+151) {
		tmp = x;
	} else if (y <= 1.55e-60) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.55e+151:
		tmp = x
	elif y <= 1.55e-60:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.55e+151)
		tmp = x;
	elseif (y <= 1.55e-60)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.55e+151)
		tmp = x;
	elseif (y <= 1.55e-60)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.55e+151], x, If[LessEqual[y, 1.55e-60], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5500000000000001e151 or 1.54999999999999994e-60 < y

   1. Initial program 90.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 90.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 77.2%

      \[\leadsto \color{blue}{x} \]

   if -1.5500000000000001e151 < y < 1.54999999999999994e-60

   1. Initial program 86.3%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 86.3%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{y}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 68.5%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{y}\right) \]

   7. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.9% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\frac{1 + y \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1000.0) (/ (+ 1.0 (* y x)) y) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1000.0) {
		tmp = (1.0 + (y * x)) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1000.0d0)) then
        tmp = (1.0d0 + (y * x)) / y
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1000.0) {
		tmp = (1.0 + (y * x)) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1000.0:
		tmp = (1.0 + (y * x)) / y
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1000.0)
		tmp = Float64(Float64(1.0 + Float64(y * x)) / y);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1000.0)
		tmp = (1.0 + (y * x)) / y;
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1000.0], N[(N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 55.7%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 55.7%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 55.7%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

      2. /-lowering-/.f64 43.1%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   7. Simplified 43.1%

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot y}{y}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + x \cdot y\right), \color{blue}{y}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot y\right)\right), y\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot x\right)\right), y\right) \]

      4. *-lowering-*.f64 56.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, x\right)\right), y\right) \]

   10. Simplified 56.8%

       \[\leadsto \color{blue}{\frac{1 + y \cdot x}{y}} \]

   if -1e3 < z

   1. Initial program 94.8%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   2. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 94.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 94.8%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

      2. /-lowering-/.f64 94.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   7. Simplified 94.8%

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.2% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (/ 1.0 y)))

C

double code(double x, double y, double z) {
	return x + (1.0 / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / y)
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / y);
}

Python

def code(x, y, z):
	return x + (1.0 / y)

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / y);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.5%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
2. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

   4. exp-to-pow N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

   8. +-lowering-+.f64 88.5%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

3. Simplified 88.5%

   \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \frac{1}{y}} \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

   2. /-lowering-/.f64 86.5%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

7. Simplified 86.5%

   \[\leadsto \color{blue}{x + \frac{1}{y}} \]
8. Add Preprocessing

Alternative 7: 49.2% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 88.5%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
2. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

   4. exp-to-pow N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

   8. +-lowering-+.f64 88.5%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

3. Simplified 88.5%

   \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
6. Step-by-step derivation

7. Simplified 49.9%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Developer Target 1: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< (/ y (+ z y)) 7.11541576e-315)
   (+ x (/ (exp (/ -1.0 z)) y))
   (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y / (z + y)) < 7.11541576e-315) {
		tmp = x + (exp((-1.0 / z)) / y);
	} else {
		tmp = x + (exp(log(pow((y / (y + z)), y))) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y / (z + y)) < 7.11541576d-315) then
        tmp = x + (exp(((-1.0d0) / z)) / y)
    else
        tmp = x + (exp(log(((y / (y + z)) ** y))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y / (z + y)) < 7.11541576e-315) {
		tmp = x + (Math.exp((-1.0 / z)) / y);
	} else {
		tmp = x + (Math.exp(Math.log(Math.pow((y / (y + z)), y))) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y / (z + y)) < 7.11541576e-315:
		tmp = x + (math.exp((-1.0 / z)) / y)
	else:
		tmp = x + (math.exp(math.log(math.pow((y / (y + z)), y))) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y / Float64(z + y)) < 7.11541576e-315)
		tmp = Float64(x + Float64(exp(Float64(-1.0 / z)) / y));
	else
		tmp = Float64(x + Float64(exp(log((Float64(y / Float64(y + z)) ^ y))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y / (z + y)) < 7.11541576e-315)
		tmp = x + (exp((-1.0 / z)) / y);
	else
		tmp = x + (exp(log(((y / (y + z)) ^ y))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision], 7.11541576e-315], N[(x + N[(N[Exp[N[(-1.0 / z), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Exp[N[Log[N[Power[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024152 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ y (+ z y)) 17788539399477/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))